Tuesday, March 24, 2015

Chords and Secants and Tangents, oh my!

Once you get to circles, chords, tangents, secants, arcs, and angles in Geometry, it's all just a nightmare of things to memorize — which means things to forget or confuse.

Rather than have students memorize this mishmash of formulas, I decided to have them focus on recognizing situations and relationships. And a good way to do that is with a four-door foldable.

Make it and use it.

Humans are tool-using animals. So let's do this thing!


Doors #1, 2, 3, and 4 each have only a diagram on their front door with color-coding because, as my brilliant colleague Alex Wilson always says, "Color tells the story."

Inside the door, we wrote as little as we possibly could. At the top we wrote our own description of the situation depicted on the door of the foldable. What's the situation? Intersecting chords. What is significant? Angle measures and arc lengths. What's the relationship? Color-code the formula.

Move on to Door #2.



We did the same thing with the chord, secant, and tangent segment theorems.




These two foldables are the only tools students can use (plus a calculator) for all of the investigations we have done with this section. They are learning to use it like a field guide — a field guide to angles and arcs, chords, secants, tangents. I hear conversation snippets like, "No, that can't be right because this is an intersecting secant and tangent situation."

My favorite is the "two tangents from an external point" set-up, which we have dubbed the "party hat situation" in which n = m (all tangents from an external point being congruent).

A big part of the success of this activity seems to come from teaching students how to make tools and to advocate for their own learning. Help them make their tools, set them loose, and get out of their way.

That's a pretty good strategy for teaching anything, I think.

Saturday, February 28, 2015

Desmos plus INBs — Conic Sections Edition

One of the things I have always been frustrated with is the crappy way example graphs look in student notebooks.

Well, no more.

For my conic sections notes sessions in Precalculus, I'm using Desmos-created graphs with all their equational and slider glory.

Here's how I ensure kids have readable, meaningful examples in their INBs:

I've created some modified graphs of the Desmos parabola graphs — one with a vertical axis and one with a horizontal axis.

I take a screen shot of the equation drawer PLUS the graph and paste it into an Omni Graffle document. For those of you playing our home game on the Mac, that's:

  • Press Cmd-Shift-4 to enter screen grab mode
  • Select the region of the Desmos window that you want to use as your graphic (this pastes it directly onto the Mac OS X clipboard)
  • Paste into a blank Omni Graffle document (from omnigroup.com )
  • Resize to fit your needs, then
  • Select, copy, and paste as many times as you need to create the master for your tiny handout
I arrange them 3-UP on the photocopy master so the tiny handout will fit onto a standards notebook/INB page.

Here are the files for my photocopy masters:
Chop, glue, annotate.

I recognize this is totally old school, but everything old is new again.

Thursday, January 29, 2015

Swan-style matching task — matching polar graphs with polar equations (Precalculus)

This was a winner. By turning a perfunctory matching activity into a group task, I created a rich mathematical investigation that achieved in one period what two days of lecture and guided practice failed to come close to.

I just grabbed these graphs and their equations from our textbook, enlarged them using Omni Graffle, and provided the markers, scissors, glue sticks, and half-sheets of poster paper.

A motivated class did the rest.

We followed with a SILENT gallery walk and a whole-class debrief. I told them to find the best ideas they could adopt from other table's posters.

It was not until I brought these into the Math Office and laid them out on the floor that several of us noticed the most original, most insightful, and deepest conceptual learning aspect of poster number 3 below, which is left as an exercise to the reader.  :)

Here's the activity file (on the Math Teacher Wiki).

And here are three of the posters the class created. Click on the photos to zoom.

Poster 1  



Poster 2
















Poster 3

What do you notice?








Tuesday, January 20, 2015

Here's an example: how I use Talking Points both before and *for* mathematical conversation

OK, here's an example of how I used Talking Points first to get students primed for listening and considering other viewpoints, and then to get them to listen to and consider other viewpoints that can cause them to change their minds.

As our first activity following our first test of the semester, we did these Talking Points to start class.


These talking points were not especially successful, but they opened the door for the similar triangles discussion that followed.

We debriefed a bit, then I handed out this lovely, subtle activity from Park Math (Book 3, #20), and I asked them to change (a) to become a Talking Point, as in, "Triangle PRQ is similar to triangle STU." They were, as always, charged with doing three rounds and justifying their opinions.


Ten minutes of conversation ensued.

Next, I wrote three headings on the whiteboard (Agree, Disagree, Unsure) and asked each table in turn to tell me which conclusion they had come to and why. One by one, I wrote the table numbers under the categories where they located themselves (Agree, Disagree, Unsure).

And I held my tongue as table after table disregarded the order of vertices to tell me that, Duh, of course, they are similar triangles. I held my tongue because I trusted the process and had a felt sense that in a room full of 37 people, surely SOMEBODY would express a different, correct opinion.

And lo, it came to pass.

Table 6 bravely offered their belief that the triangles named were not similar because the order of vertices in each was not corresponding.

And one by one, the little lightbulb moments popped around the room.

I kept the discussion going until we were through with all 9 tables. Then, and only then, did I give tables another round in which they could change their opinion about what was actually going on in the diagram. 

Afterwards, we discussed what had happened. What did happen, I asked them. And they responded that something they heard made them realize they wanted to change their minds.

So that was my perfectly imperfect day of Talking Points. On the one hand, kids understood (some for the first time) that listening to somebody else could have value for them. On the other hand, many spent most of the exercise not listening to each other and simply waiting for their own turn to talk.

This doesn't mean that it was a failure. It just means it was a first step. 

I believe that if you want students to take ownership of their own learning (and listening... and opinions), then you have to allow space for them to do it in their own perfectly imperfect way. I have found that when I trust the process, I get the best results.

I am posting this to help you understand that every round of Talking Points I do is not a cornucopia of unicorns and rainbows.

Monday, January 19, 2015

On Talking Points, disidentification, meditation, and the need for a structure

One of the things I have noticed with discouraged math learners and Talking Points — or other disidentification techniques — is that students often express a kind of euphoria afterwards. “That was so much fun!” they will often say (or yell).

This is a result of the disidentification process. They are not accustomed to speaking in their own authentic voices in math class. They have become conditioned to attending math class under what Brousseau called “the didactic contract.” Under the didactic contract — the implied contract to which they have become conditioned — they are required to check their authentic self at the door, with all its attendant messiness, blurting-out, hesitant and half-formed ideas. Instead, the didactic contract demands that they conform to very narrow ideas of what a “good math student” is supposed to do: Sit down. Shut up. Pay attention. Get the right answer. Don’t ask why.

There’s no point in our denying that this is what most math students have become acculturated to. They didn’t make up these requirements themselves. Somewhere along the way, everybody encounters a math learning environment in which these are the expectations.

This situation is what the great Swiss psychoanalyst Alice Miller spoke of in her book, The Drama of the Gifted Child. In an adult-centered, authoritarian society (which most societies are), the social constructs are organized to side always with the adult/parent/teacher instead of with the child. And in fact, as Miller’s work showed, not just *instead of* the child, but at the child’s expense.

This is the general situation for discouraged learners in math classes. Many students perceive early on that their authentic selves are not welcome here. They quickly learn to wear a mask in math class and to pretend to be smart, compliant, and “mathematical” — in other words, to adopt a false persona in math class.

The problem for us as math teachers is that it is not easy to crack through this false self. Disidentification is not an easy or a straightforward process. The psyche adopts these masks as defense mechanisms — and for very good, if outdated, psychological and emotional reasons: fear of abandonment, fear of humiliation, fear of shaming, fear of annihilation.

These may be outdated, outmoded fears by the time a student reaches middle school or high school, but that does not make them any less real or active in the present moment. For a traumatized math learner, they are the most real thing in their world during 5th period.

In their own different ways, the psychologists Eugene Gendlin, A.H. Almaas, and Francine Shapiro have all posited that trauma in the past forms a kind of “stuck place” in the human mind/brain/psyche. Whenever we encounter a stimulus that “triggers” that stuck place, we “flash back” to the moment of trauma and our defense mechanisms lock into place.

This is what makes disidentification so difficult to achieve in practice. A defended psyche is not a receptive psyche. And a student may *hear* that s/he needs to adopt a growth mindset in math class, but s/he hears this message from his or her bunker, thirty feet under ground and behind several feet of concrete protective functions.

Raise periscope. Spot the threat. Lower periscope and retreat.

Evolutionary psychologists consider this fight-flight-freeze response and its replay during anxiety dreams as a most ancient form of threat rehearsal. Knowing what they know from their previous experience, the protective functions of the psyche leap into action and do their best to make sure we remain vigilant and safe from incoming threats. They perceive this to be a matter of survival, which is why they go to such great lengths to make sure we perceive it that way too whenever we step over the threshhold into math class.

So the first order of business in the process of disidentification is to establish trust and to form a safe — and sane — alliance with all learners. If math class is to become a growth mindset place for all students, then it must first be established as a safe place in which to remove our masks and to return to being our deeper, authentic, creative selves.

To make any place safe for the authentic self to come out, it helps to have a structure in place. That way, the structure can provide the psychological and emotional safety (and freedom) in which we can drop down into our authentic selves.

In all forms of mindfulness meditation, this structure consists of three things: a posture, an anchor, and a timed period.

In Zen, we sit on a black cushion in the lotus or half-lotus position (or forward on a chair with both feet flat on the floor). We place our hands on our knees or in the cosmic mudra and we face a white wall. We lower our gaze to a 45-degree angle with the floor, and we anchor our attention on our breath.

Whenever our attention wanders — and monkey mind guarantees that it will inevitably wander — we gently redirect it back to our breathing.

The Vietnamese Zen teacher Thich Nhat Hanh teaches the use of a gatha, or mindfulness verse, as an attentional aid during meditation. With each in-breath or out-breath, one thinks a line of a simple verse:

Breathing in, I calm my mind.
Breathing out, I smile.
Dwelling in the present moment,
I know that this is a wonderful moment.

Which reduces to:

In,
Out.
Present moment,
Wonderful moment.

I’ll say this about Thich Nhat Hanh: you have to be a pretty evolved being to be able to teach this kind of clarity and sanity to the very countries that launched your own into chaos.

We do all of this for the whole timed period, whether it is ten minutes or 45 or an hour. Gradually, with patience and lovingkindness, we learn how to do this for longer and longer periods, until the timed period we are working with is every day for the rest of our lives.

We do this because this is our structure.

To the uninitiated, a structure might seem to be a rigid thing, but that is a misunderstanding, and I will tell you the secret: it is actually the essence of freedom.

It gives our defense mechanisms and our wounded child ego-self-psyche something important to do while we drop down into the vulnerable place where our authentic self is kept safe — beneath all those layers of protective functions, social masks, people-pleasing, snark, and our “on-stage” personas.

The structure makes it safe for a human being to reconnect with that deeper, authentic self.

So it is natural to experience a kind of euphoria afterwards. Our culture generally doesn’t encourage us to connect with our authentic selves, so when we do, many people experience it as a kind of homecoming. Intuitively, we know that it is the source of all our greatest ideas and energy and creative fire. Finally, it is a relief to drop the masks we wear and to just be fully and authentically ourselves.

The Enlightenment poet Friedrich Schiller described this experience of flow as arising from the competing impulses toward being present and toward thinking, which operate in a kind of luminous reciprocity, with their harmonious interaction producing a third impulse which he terms the Spieltrieb (or 'play impulse'):

Irresistibly seized and attracted by the one quality, and held at a distance by the other, we find ourselves at the same time in a condition of utter rest and extreme movement, and the result is that wonderful emotion for which reason has no conception and language no name.
                       — Friedrich Schiller, Twelfth Letter on the Aesthetic Education of Man

When the mind is both fully at play and fully at rest in this way, it is at home. 

And when this experience happens in math class, students are growing and truly experiencing mathematics.

This is the sanest, healthiest, richest, most creative human state I know — and I want all of my students to experience it in my math class. Only then can they connect with the growth mindset and the mathematics that are their birthright.

But the key to unlocking that moment is through structure. And for me, in my math classes, that structure is Talking Points.

Friday, January 16, 2015

New Discoveries About Talking Points (BRIEF POST)

Through a conversation with Ilana Horn (@tchmathculture) and Michael Pershan (@mpershan) this morning at 5 am Pacific Time, I've had another insight about why I think that Talking Points is such a powerful activity structure: it is a non-coercive way for students to "try on" different ideas about mindset (or anything else) during their time of greatest neuroplasticity.

I cannot tell you how many times during Talking Points I've heard a student say, "I never thought of it that way before, but I kind of agree with the statement because ____..." (side note: justification and listening are the two most important habits of mind that Talking Points help to cultivate).

I cannot tell you because it has happened so many more times than I can count.

This is related, I suspect, to something else I've been noticing lately — that there are a lot of people who believe in their hearts that they are implementing Complex Instruction, but in fact, they are just using the structures of Complex Instructions as tools for implementing a coercive classroom management strategy.

To me, this is at odds with my goals for social justice, equity, and inclusion, which means that I really have to walk the talk in my classroom every day.

More soon but I wanted to capture these thoughts.

Friday, October 24, 2014

On the importance of scaffolding within the zone of proximal development

I don't know about you, but my experiences with even the best Shell Centre tasks (their Formative Assessment Lessons, or "FALs") have been hit or miss. These are among the best anywhere, and while I generally love the ideas of their tasks, in practice, I almost always need to tweak their implementations to make them work in my classroom and with my students.

I wonder if this is not inevitable, given the highly customized nature of scaffolding.

I experienced this phenomenon yet again today when I used their Ferris Wheel task, which I brought out for my extremely able but easily discouraged Precalculus students:

    http://map.mathshell.org/materials/download.php?fileid=1252

The purpose of this task is to jump-start students' understanding of modeling and graphing trigonometric functions, using the movement of a ferris wheel cart together with some scaffolded information. The pre-assessment task scaffolded the process well, except for the crazy way they laid out the equations. But then when I ran the card sort task in my 1st period classroom, it was a complete trainwreck. The changes in set-up seemed to yank the problem out of the zone of proximal understanding and launched it into the stratosphere. When something is new, my students are easily thrown (and discouraged), so when they shifted from providing the period to providing the number of revolutions in some arbitrary number of minutes, my students' heads exploded. There was a domino effect of cascading fear and consequences — the equations are set up strangely and are given in degrees rather than radians (after all our hectoring!), the vocabulary changed from revolutions to rotations, etc etc etc.

So during my prep, I made new versions of the cards — ones that would allow my 5th and 7th period classes to recognize the new material and to connect it to things that were still strange, but at least strange in ways they could notice and recognize.

The differences were dramatic. 1st period groups were upset because they were unable to make what they considered meaningful progress on the task (in spite of excellent mathematical conversations) and left feeling unsteady and discouraged. Here is the work from an especially confident and capable 1st period group:



My two later classes, on the other hand, were able to attack the task, making meaningful connections, spot patterns, and really solidify their understanding. Here is the work from an often-cautious and un-confident (but still very capable) end-of-day group:



All in all, it was a good reminder to me to avoid changing horses in mid-stream when I am trying to help learners build conceptual understanding through good scaffolding!